generalized maxwell−stefan approach for swelling kinetics of dextran gels

Generalized Maxwell-Stefan Approach for Swelling Kinetics ofDextran Gels

M. A. T. Bisschops, K. Ch. A. M. Luyben, and L. A. M. van der Wielen*

Kluyver Institute for Biotechnology, Delft University of Technology, Julianalaan 67,2628 BC Delft, The Netherlands

The swelling kinetics of two types of Sephadex dextran gels in water is described by a modelbased on the generalized Maxwell-Stefan (GMS) relations. The driving force for the swellingprocess is the gradient of the chemical potential gradient, which includes terms for mixing ofsolvent and polymer as well as for the elasticity of the swollen gel network. The friction isrelated to the relative motion of the components, via an effective diffusion coefficient. This isdescribed as a function of the volume fraction of polymer via the Ogston relation (diffusion based)and a hydrodynamic model (based on the analogy with viscous flow). The resulting swellingmodel is evaluated via rigorous solution of the equations (homogeneous driving force approach,HDF), in which intraparticle gradients are taken into account, and approximated by a single-step central difference method (linear driving force approach, LDF). The latter allows muchfaster evaluation of the swelling process. These two numerical solution methods give similarresults for the simulation of the expanding outer radius of the gel particles, regardless of whichexpression is used for the effective diffusivity. When intraparticle dynamics are considered (HDFsolution method), some differences between the two effective diffusivity relations arise. Thehydrodynamic model predicts steeper composition gradients than the Ogston model and leadsto a distinctly later dissolution of the shrinking core. However, there is a lack of reliable dynamicintraparticle profile data to verify either model.

Introduction

Hydrogels are cross-linked hydrophilic polymers, usu-ally with a low degree of cross-linking. Hydrogels canabsorb significant amounts of water to form elastic gels.They have found a wide range of applications, includinguse in electrophoresis, immobilization of enzymes, andcontrolled release of pharmaceuticals, use in biosepa-rations, and use for moisture retention in soil.1

Swelling equilibria of gels have been studied exten-sively, both theoretically and experimentally,2-6 andexcellent reviews on the thermodynamic model ap-proaches are available.7 For some of the applicationsmentioned above, knowledge on the swelling kineticsis strongly desired, but the number of studies in thisfield is limited.

In the few quantitative studies on swelling dynamics,the results are usually reported as swelling curves, inwhich either the particle volume8,9 or the particle radiusor diameter is plotted as a function of time.10-13 Whenthe experiments are carried out in a single solvent, theswelling curves show a common pattern, irrespective ofthe type of hydrogel. In all cases, a rapid initial swellingis observed, followed by a slower approach to the finalequilibrium state. Similar results are reported forsystems in which low concentrations of solutes arepresent. In experiments with higher concentrations ofsolutes, the bead radius can show a maximum, followedby a gradual decrease toward the equilibrium state.11

Very little attention has been paid to the intraparticleswelling behavior. Lee11 investigated the release ofthiamine hydrochloride from poly(2-hydroxyethyl meth-acrylate) (PHEMA) beads and found clear visual evi-

dence that the swelling process exhibits shrinking-corebehavior, in which a solvent front gradually penetratesthe particle. This solvent front separates an outerswollen region from a glassy core. At approximately50-80% of maximum swelling the solvent fronts meetand the shrinking core vanishes.

A rigorously correct model should relate the observedswelling dynamics to molecular properties and interac-tion parameters. Such a model should be able todescribe the behavior of the hydrogel from its initial dryglassy state to its fully swollen elastic state, includingthe observed shrinking-core behavior.

Vavruch14 presented a quantitative description of theswelling kinetics of dextran hydrogels. This model isbased on first-order kinetics, in which the swelling rateis proportional to the difference between the prevailingsolvent content of the gel and the solvent content in theequilibrium state. This approach was later adopted byPratt and Cooney8,9 who fitted this model to swellingexperiments with two types of dextran gels of differentsize and cross-linking in water and aqueous proteinsolutions. Their model could describe the experimentaldata reasonably well, although a closer examinationindicates a systematic underprediction of the initialswelling rate (until approximately 50-70% of themaximum particle size) and an overprediction of theswelling rate beyond that point. The first-order kineticmodel requires a single parameter, which stronglydepended on particle size, the type of gel, and thesolvent. This parameter, therefore, is not solely acomponent property or a molecular interaction param-eter.

Tanaka and Fillmore10 developed a model in whichthe swelling process is described as a balance betweena driving force and friction. The driving force is as-

* To whom correspondence should be addressed. Fax: +3115 278 2355. E-mail: [email protected].

3312 Ind. Eng. Chem. Res. 1998, 37, 3312-3322

S0888-5885(98)00038-4 CCC: $15.00 © 1998 American Chemical SocietyPublished on Web 07/17/1998

sumed to result from a uniform stress that is latent inthe unswollen network. Tanaka and Fillmore10 referto this stress as an osmotic pressure. When the networkis transferred into a fluid, this pressure forces thenetwork to expand until the osmotic pressure of the gelbecomes equal to zero. The friction between the net-work and solvent is described by means of a constantdiffusion coefficient, which is related to the bulk andshear modulus of the polymer network.

Tanaka and Fillmore derived an analytical expressionfor their model, which shows remarkable resemblancewith the equation for diffusion in nonswelling sphericalparticles in an infinite batch.15 As a consequence, thepredicted swelling curve and intraparticle concentrationprofiles show behavior similar to the concentrationprofiles according to Fick’s law.

Komori and Sakamoto16 criticized the approach ofTanaka and Fillmore, arguing that the driving force forthe swelling process cannot result from a stress that islatent in the network before it is contacted with thefluid, because the unswollen network is mechanicallystable in itself. Furthermore, Komori and Sakamotodisagree with the proposed mechanism for friction, andthey indicate that the quantitative relation between thediffusion coefficient and bulk and shear moduli asproposed by Tanaka and Fillmore cannot hold.

Komori et al.12 presented another approach in whichthe driving force for the swelling process is assumed toresult from a gradient in the chemical potential of thesolvent, expressed as a balance between the swellingpressure and the bulk modulus of the network. Thefriction is described by an ordinary diffusion equationwith a constant diffusion coefficient. Although themodel is based upon a different physical reasoning, theresulting mathematical expression of this model isanalytically identical with the Tanaka-Fillmore model.

This model shows reasonable correspondence withexperimental data but again systematically underpre-dicts the initial swelling rate.12 Moreover, Tanaka andFillmore already acknowledge that their model is onlyapplicable to linear systems, in which the Hookeanelasticity relationship is valid. For large swelling rates,this assumption may no longer hold. At last, Komoriet al.12 found that the diffusion coefficient of water insodium acrylate gels increased significantly with in-creasing the degree of cross-linking. This anomalousbehavior was attributed to stronger energetic interac-tions between the solvent and polymer in the case ofdenser gel networks. However, if the diffusion coef-ficient is considered to represent friction between thesolvent and polymer segments, this explanation cannothold and the diffusion coefficient should be almostindependent of the degree of cross-linking.

None of these models predicts the shrinking-corebehavior observed by Lee. The first-order kinetics doesnot contain any information on intraparticle phenom-ena, and the Tanaka-Fillmore model predicts concen-tration profiles similar to the Fickian diffusion equationderived by Crank.15 They, therefore, fail to predictshrinking-core behavior. Moreover, the current modelscannot be straightforwardly extended to multicompo-nent systems, which include cosolvents and higherconcentrations of solutes. Therefore, in this work, thegeneralized Maxwell-Stefan (GMS) relation17 is used.It is based on an equilibrium between driving forces formass transfer and the friction forces between thevarious components involved. The net driving force is

the combination of entropic and enthalpic effects of thepolymer-solvent interaction and the network elasticity.

The aim of this paper is to demonstrate the potentialsof the generalized Maxwell-Stefan approach for de-scribing the swelling kinetics of hydrogels. Therefore,the volumetric change of the gel particle as well as theintraparticle shrinking-core behavior is considered quali-tatively and quantitatively. This model is verified withexperimental data of Pratt and Cooney.8,9

Generalized Maxwell-Stefan Equation

The GMS relation reads as

in which Fdr,i represents the net driving force on the keycomponent and Ffr,i the friction between the key com-ponent (i) and all other components.

The net driving force is due to the gradient in thechemical potential of species i.17 The expression for thechemical potential of components in a system consistingof a hydrogel and a solvent is determined by threecontributions, which are additive: (1) the entropiccontribution, which accounts for mixing of the polymernetwork and the solvent, (2) the enthalpic contribution,which accounts for the interactions between the networkand the solvent molecules, and (3) the elastic contribu-tion of the network, which represents the work requiredfor expanding the network, which leads to a decreasein entropy of the polymer chains.

The chemical potential of the solvent in a mixture ofnon-cross-linked polymer chains and a solvent is de-scribed by the Flory-Huggins theory.18 For a binarymixture, which consists of polymer and solvent withoutsolutes and cosolvents, the chemical potential of thesolvent reads as

where φ represents the volume fraction of polymer inthe gel and ø is the usual Flory-Huggins parameter,which is predominantly determined by the enthalpiccontribution. The reference state (µs

0) is convenientlychosen as that of the pure solvent in the liquid state.

The classical Flory-Huggins theory is based on arandom-mixing lattice model and can successfully de-scribe phase equilibria for nonpolar polymers in non-polar solvents. However, random-mixing models do notaccount for orientation-dependent interactions, such ashydrogen bonds. These interactions could influence thebehavior of aqueous systems.19

Nevertheless, since the Flory-Huggins model is themost widespread model used for solvent-polymer in-teractions, it has also been used extensively for systemsin which hydrogen bonds are present. The fitted Flory-Huggins parameter in such a system no longer solelyaccounts for enthalpic effects but may also incorporateentropic elements, such as the molecular weight of thepolymer. Moreover, the temperature dependence of themodel can no longer describe nonisothermal processescorrectly.19

The objective of this paper is to develop a generalmodel that describes swelling under isothermal condi-tions. Therefore, the Flory-Huggins mixing theory will

Fdr,i ) Ffr,i (1)

µs – µs = RT(ln(1 – φ) + φ + χφ2)0

entropiccontribution

enthalpiccontribution

(2)

Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3313

be implemented, using the parameter fitted at thecorrect temperature.

The elasticity term originates from the change inentropy upon stretching the polymer network. It isexpressed as the derivative of the Helmholtz energy ofthe hydrogel with respect to the volume.18 This volumederivative is related to the extent of deformation of thepolymer chains. In the ideal case, there is no changein internal energy when a polymer chain is deformed.Hence, the elasticity contribution represents a purelyentropic phenomenon.

The degree of freedom in the network depends on thedegree of cross-linking and on the behavior of thejunction points. Common models that describe thisbehavior are the affine network model20 and the phan-tom network model.21 The former assumes fixed junc-tion points, whereas the latter allows the junction pointsto move freely in space. Qualitatively, these modelsshow the same swelling behavior, but the phantomnetwork model performs slightly better at high swellingratios.19 In this work, we have used the phantomnetwork model. The elastic contribution to the drivingforce for the swelling process is described by

where νS represents the partial molar volume of thesolvent, Fn is the density of the polymer network in drysolid state, and Mc represents the average molecularweight between two junctions in a polymer chain. Thenetwork functionality (Φ) at a cross-link usually has avalue of 3 or 4, depending on the type of polymer andcross-linking agent.

The net driving force for the swelling process is thegradient in the combined potential energy, which is thesum of the Flory-Huggins chemical potential and theelastic term:

Note that the driving force has the required dimensionof N/mol.

Friction between the Solvent and Polymer. Thefriction force exerted on a component in the system isrelated to the velocity of that particular componentrelative to the velocities of other species in the system.In the case of swelling in a single solvent, only twospecies are involved, the polymer and the solvent, soonly one velocity difference is involved. The frictionforce that is exerted by component j on component i isproportional to the number of molecules or the molefraction of the first species. In the case of cross-linkedmacromolecules, these quantities are probably inap-propriate due to the large size differences and the cross-links. Therefore, the friction force acting on the solventis assumed to be proportional to the volume fraction ofthe polymer in the gel phase. The expression for frictionon the solvent molecules now reads as

The friction between small molecules and porous media,such as polymer networks, has some peculiarities.First, due to steric hindrance, the pathway through themedium is longer than the distance in a straight line,

which leads to increased friction and a lower effectivegradient. Moreover, narrow pores lead to an additionalincrease of the friction due to the smaller solute-to-porediameter ratio (constrictivity). These effects are in-cluded in the effective diffusion coefficient (Deff), whichthus depends on the degree of swelling:

where D0 represents the diffusion coefficient of thespecies in free solution. Several expressions have beenproposed for this functionality.22 In this work, two typesof functions are considered.

(a) Ogston Model. Ogston23 derived a model on thebasis of the stochastic description of diffusion, originallyproposed by Einstein. This model takes into accountthe free space available in the swollen network. Thenetwork is assumed to consist of a random array ofuniform fibers. Ogston applied this approach success-fully in an equilibrium model to predict partitioningbehavior of solutes in hydrogel-water systems as well.24

The effective diffusion coefficient of a (spherical) mol-ecule in such a network can be described by

in which λ represents the ratio between the radius ofthe polymer chain and the diffusing molecule (λ ) rP/rS). Vonk22 investigated the application of the GMSapproach for mass-transfer phenomena in various hy-drogels. He found that the experimental data could verywell be described with a slightly modified expression,in which the exponential term is proportional to λdirectly rather than to λ + 1.

(b) Hydrodynamic Model. A different approach isbased on the analogy between diffusion of solventsthrough a porous matrix and viscous flow through a bedof particles or fibers. The latter can be described usingDarcy’s law:25

The left-hand side of the equation corresponds to thedriving force, which in this case is solely composed of apressure gradient. The friction term is, as mentionedbefore, proportional to the velocity difference, the vis-cosity of the solvent (η), and the reciprocal of thepermeability of the fiber bed (B). The latter can bedescribed using the well-known Kozeny-Carman rela-tion, which reads as follows for a bed of fibers:

in which ε represents the void fraction, which is definedas the volume fraction of liquid in the fiber bed, arepresents the specific interfacial area per unit volumeof the fiber bed, df represents the fiber diameter, andK0 represents the Kozeny constant. Usually K0 ) 5gives reasonably accurate values.25

Combining these equations yields

If the analogy between the hydrodynamics of viscous

µel ) νSRT[ Fn

Mc(1 - 2

Φ)( φ

φ0)1/3] (3)

Fdr ) - ddr

(µFH + µel) (4)

Ffr ) (RTDeff

)φ(uS - uP) (5)

Deff ) D0f(φ) (6)

Deff/D0 ) exp(-(1 + λ)xφ) (7)

-dP/dz ) (η/B)(uS - uP) (8)

B ) ε2

K0a2

)df

2

80( ε2

(1 - ε)2) (9)

80ηdf

2

(1 - ε)2

ε2

(uS - uP) ) - dPdz

(10)

3314 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998

flow through a bed of fibers at macroscopic scale anddiffusion of solvent inside a gel bead at molecular scaleholds, the effective diffusion coefficient should have thefollowing form:

Wesselingh and Krishna17 obtained a similar expressionfor gas permeation through a membrane, in which themembrane is considered to consist of a bed of spheres.

The main disadvantage of eq 11 is that it does notmeet the requirement that the effective diffusion coef-ficient equals the diffusion coefficient in an infinitelydiluted polymer solution (f(φ) * 1 at φ ) 0). Thisdisadvantage is also acknowledged in the Kozeny-Carman relation for fiber beds with high void frac-tions.25 This is usually accounted for by increasing theKozeny constant (K0). For silk fibers values up to K0 )15 have been reported.

Bootstrap Relation. The swelling process involvesmotion of at least two species: the polymer networkmoving outward and the solvent molecules movinginward. The GMS relation only describes the relativemotion of these components. The additional relationthat is required to relate the relative motion to absolutefluxes is known as the “bootstrap relation”.17

In this case, the bootstrap relation follows from thecontinuity equation. Assuming both components to beincompressible, this is more conveniently expressed involumes. At any distance from the center of the bead,the volumetric flux of the solvent equals the volumetriccounterflux of the polymer chains. The solvent fluxequals the product of linear velocity of the solvent (uS)and the volume fraction of solvent (1 - φ), and the fluxof polymer chains equals the product of the linearvelocity of the polymer (uP) network and the volumefraction of polymer (φ). The mass balance is nowwritten in terms of volumetric fluxes (or velocities):

The linear velocity of the polymer at any distancefrom the center of the gel particle equals the velocity atwhich the polymer network travels outward at thatparticular radius. Since the linear velocities of thepolymer network and the solvent are related to thefluxes and the polymer volume fraction, the slip velocityat any distance from the center of the bead is expressedin terms of the local swelling velocity:

Therefore, the linear velocity at which the outsideelement in the polymer network moves outward equalsthe observed swelling velocity of the hydrogel particle.

Mathematical Solution of the GMS Equation.The complete GMS equation for swelling of hydrogelsis now obtained by substituting the expressions for thedriving force and friction into the general GMS equation(eq 1):

The bootstrap relates this GMS equation in terms of thelinear velocity at which the polymer network travels

outward at any position in the gel bead. The result iswritten in dimensionless form as

where θ ) tD0/r02 represents a dimensionless time,

analogous to the Fourier time, and x ) r/r0 representsthe dimensionless radial coordinate.

The dimensionless radial coordinate and the dimen-sionless time can be based on any reference radius.Contrary to the work of Tanaka and Fillmore,10 we willexpress these dimensionless numbers relative to theinitial (dry) gel bead radius (r0). The final (equilibrium)particle radius depends on the process conditions, suchas the type of solvent and the presence of solutes, andis not necessarily known beforehand. The consequenceof defining the dimensionless numbers relative to theinitial bead radius is that the dimensionless radiusexceeds unity because the particle in its swollen stateis larger than that in its initial dry state.

The GMS relation derived above is a nonlineardifferential equation that has to be solved numerically.The boundary condition is that the chemical potentialof the solvent at the outside radius in the gel beadequals the chemical potential in the surrounding liquid:

In the case of swelling in contact with a pure solvent,the chemical potential of the solvent in the liquid phaseis chosen equal to the reference state (µs

L ) µs0). The

initial condition depends on the initial state of the gelbead. For a dry gel particle, this would be

(a) Homogeneous Driving Force Approach. Thehomogeneous driving force (HDF) approach takes localswelling phenomena inside the gel bead into account.Therefore, it involves evaluation of the local drivingforce at any position inside the gel bead. An analyticalsolution of the nonlinear differential equation appearsimpossible, and a numerical procedure is followed. Thisinvolves discretization of the particle in radial concentricelements and solving of the resulting set of differentialequations in time.

The radial discretization scheme that will be used inthe development of the numerical procedure is shownin Figure 1. The polymer matrix is subdivided intosegments, which will move outward at a certain velocity(uP). The velocity at which the outside segment willtravel outward, of course, corresponds to the swellingrate. The velocity at which each segment travels nowis

where φi/ represents the volume fraction of polymer on

the boundary between cells i and i + 1, which isobtained by linearly averaging the volume fraction ofpolymers in these cells. The distance over which thepotential gradient is evaluated (δxi) is taken as thedistance between the centers of elements i and i + 1.The position of each segment follows from the integral:

f(φ) )Deff

D0∝

(1 - φ)2

φ(11)

uS(1 - φ) + uPφ ) 0 (12)

uS - uP ) -( 11 - φ)uP (13)

(RTD0

) 1f(φ)

φ(uS - uP) ) - ddr

(µFH + µel) (14)

uP(x) ) (∂x∂θ)

x) f(φ) (1 - φ

φ ) ∂

∂x (µFH + µel

RT ) (15)

(µFH + µS)|r)rP) µs

L (at any θ) (16)

φ(θ)0) ) φ0 (at any r or x) (17)

uP,i ) f(φi/) (1 - φi

/

φi/ ) 1

δxi((µi+1FH + µi+1

el ) - (µiFH + µi

el )RT )

(18)


Since the amount of polymer chains in every grid cellremains constant during the swelling process, thevolume fraction of polymer chains in a cell can be relatedto the position of the boundaries of that cell:

The initial condition for the swelling process is trivial:the particle is in its dry state (φ ) φ0) at the beginningof the particle. For a grid in which all concentric cellsinitially have the same thickness (not volume), theinitial conditions with respect to position and composi-tion profile are mathematically formulated as

The outside grid cell is in direct contact with thesurrounding liquid. The polymer fraction at the inter-face between this outside grid cell and the liquid (φN

/ ) isthus assumed to be in the equilibrium state (φ∞). Thediscrete form of the swelling rate equation for theoutside grid cell thus is

in which X represents the dimensionless radius of theparticle (rP/r0), which, of course, corresponds to theoutside radial coordinate (X ) xN). The boundary at theinterface between the gel particle and the solvent (µFH

+ µel|x)1 ) 0) is thus automatically implemented in thisprocedure. The numerical calculations were done inMatLab for Windows (version 4.2).

(b) Linear Driving Force Approach. In fact, allmass-transfer phenomena that are described by theGMS relation yield a differential equation. To reducecumbersome numerical calculations, Wesselingh andKrishna17 suggest the use of a single-step centraldifference approximation, which we will refer to as thelinear driving force (LDF) approach. In most cases, thissimplification allows one to express fluxes explicitly.Wesselingh and Krishna demonstrated that this ap-proach leads to very rapid calculations with reliableresults. The main assumption in the LDF approach isthat the entire mass-transfer resistancesand conse-quently the composition gradientsis located in a (thin)film at the phase boundary. The gel phase compositionin the core of the particle is assumed to be uniform. Thissituation is schematically shown in Figure 2.

The GMS relation for the LDF approach reads as

where X is the dimensionless (total) radius of theparticle (rP/r0) and δ represents the thickness of themass-transfer film. Since the entire friction is locatedin the mass-transfer film, the friction depends on theaverage polymer volume fraction in the film (φh). Thepolymer fraction in the film is assumed to vary linearlyfrom its equilibrium state (φ∞) at the liquid interface tothe value in the core of the gel bead (φ). The averagepolymer fraction in the film is evaluated halfwaythrough the film thickness and therefore is approxi-mated by the linear average of these two:

Figure 1. Schematic representation of the numerical scheme used in evaluating the homogeneous driving force (HDF) approach for theswelling process.

xi(θ) ) xi,0 + ∫0

θuP,i dθ (19)

φi

φ0)

ri,03 - ri-1,0

3

ri3 - ri-1

3)

xi,03 - xi-1,0

3

xi3 - xi-1

3(20)

xi ) iN

(for i ) 1, 2, ..., N) and φi ) φ0 (at θ ) 0)

(21)

uP,N ) ∆X∆θ

) f(φ∞) (1 - φ∞

φ∞)( 2

xN - xN-1)(-(µN

FH + µNel )

RT )(22)

dXdθ

) f(φh) (1 - φhφh ) 1

δ/r0(µFH + µel

RT ) (23)


The driving force in the LDF approach is given by thepotential gradient over the film (∆µ/δ) and is thusrelated to the difference between the final equilibriumstate and the actual state in the core of the gel bead.The potential gradient is thus evaluated with the “bulk”polymer volume fraction in the core of the gel bead (φ).

The film thickness (δ) is taken to be proportional tothe particle diameter. This ratio between the particlediameter and the film thickness (dP/δ) is known as theSherwood number. The particle diameter increasesduring the swelling process. This volume increaseshould be taken into account in the film thickness.Since the total amount of polymer network in a gel beaddoes not change during the swelling process, the beadradius and the polymer volume fraction are related (φ0) φX3). The resulting equation is

where φ0 represents the initial polymer volume fraction.This ordinary differential equation has to be solvednumerically. The Sherwood number for mass transferinside a particle in a stagnant fluid is Sh ) 2/3π2, whichis usually rounded to Sh ) 6.6. The derivation of thesenumbers is included in the appendix. This Sherwoodnumber is only valid for longer contacting times (Fo >0.10). For very short contacting times, the penetrationtheory should be applied.

Kinetic Swelling Data

Experimental data on the swelling behavior of gelsare usually obtained by visual observation of a charac-teristic dimension of a gel particle in time. Pratt andCooney8,9 investigated the swelling behavior of two typesof Sephadex gels, the G-25 and the G-50 type gel. Thenomenclature (indicated by “G-25” and “G-50”) is basedon the amount of water that is taken up by a dry gelparticle (the water regain of Sephadex G-25 is 2.5 g/g

of dry gel; the water regain of Sephadex G-50 is 5.0 g/gof dry gel) at 293.15 K.

The reported experiments were performed in puredistilled water and in protein (BSA) solutions withvarying protein concentrations. There were no sub-stantial differences observed between the swellingkinetics in pure water and in protein solutions. Thiswas later confirmed by Monke et al.,13 who did similarwork on Sephadex G-75.

The experimental data collected by Pratt and Coon-ey8,9 were reported as swelling curves in which thedimensionless volume ((V - V0)/(V∞ - V0)) was plottedversus time. Each run covered the swelling process ofone single spherical particle. The initial particle sizeranged from 191 to 580 µm. The data sets wereclassified in four groups. The number of runs and datapoints in each experimental system is reported in Table1. The protein solutions contained 0.15 N NaCl andBSA at the following concentrations: 0.10, 1.0, and 5.0wt %.

Parameters Estimation. To allow application ofthe GMS models, the following information is required:

(a) The activity of the solvent in the surroundingliquid has to be known. In the present work, theswelling was investigated in pure distilled water, havingunit activity.

(b) Sephadex gels consist of an epichlorohydrin cross-linked dextran polymer network. The Flory-Hugginsparameter of dextran-water has been reported to bein the range of 0.49-0.51.26 Ideally, the Flory-Hugginsparameter should not vary with the degree of cross-linking. Yet, Kang and Sandler26 report a slight influ-ence on the molecular weight of the different types ofdextran. Nevertheless, we will take its value equal forboth gels at ø ) 0.50.

(c) The exact gel network properties, such as func-tionality, density, and degree of cross-linking, are themost difficult parameters to obtain. Since these havenot been reported for the two types of gels, these valueswere fitted to macroscopic observations reported byPratt and Cooney8,9 and Vonk22 and to the informationgiven in the Pharmacia product sheets. The nature ofthe epichlorohydrin cross-links indicates that the func-tionality (Φ) is approximately 4. The (solid) polymerdensity can be found by exploring the wet densities ofthe different Sephadex gels, assuming that their ma-trices are essentially identical (i.e., neglecting theinfluence of the epichlorohydrin cross-links). The wetdensity is the weighted sum of the polymer networkdensity and the solvent density in a fully swollen state.Since the water uptake of the different Sephadex gelsis known, the solid-state density can be calculated. Itwas found that the polymer density is Fn ) 1628 kg/m3

and that the volume fraction of the polymer in the drygel bead equals unity. The last parameter that ismissing is the molar mass of a chain segment betweentwo cross-links. This parameter is directly related tothe degree of cross-linking in the gel. For the two gels,this value was fitted to the equilibrium swelling ob-served experimentally. The results of the parameterestimation are shown in Table 2.

Figure 2. Schematic representation of the linear driving force(LDF) approach for the swelling process.

φh )φ∞ + φ

2(24)

dXdθ

) Sh2 ( φ

φ0)1/3(1 - φh

φh ) 1f(φh)(∆µ(φ)

RT ) (25)

Table 1. Overview of the Swelling ExperimentsCollected by Pratt8

systemno. ofruns

no. ofpoints

particle size(µm)

Sephadex G-25-water 6 189 191-407Sephadex G-50-water 6 213 261-547


Results

The swelling dynamics of both types of Sephadex gelswere simulated with both the HDF approach and theLDF approach. The effective diffusion coefficients wereevaluated by the hydrodynamic model and by theOgston model in either case.

Correspondence with Kinetic Swelling Data.The calculated swelling curves have been fitted to theexperimental data reported by Pratt8 to estimate thediffusion coefficient at infinite dilution (D0). The dif-fusion coefficients for the separate gels as well as anoverall diffusion coefficient, able to describe the swellingbehavior of both gels, have been fitted.

The estimated diffusion coefficients are given in Table3. The relative errors reported in this table are calcu-lated using

The HDF approach was evaluated with 10, 20, 40, and80 concentric grid cells in the gel particle. The time stepwas adjusted such that the maximum displacement ofany of the segments in the radius was bound to amaximum, which depended on the number of segments.The deviation in the estimated diffusion coefficientobtained from the simulations based on 40 and 80elements is below 1%.

The swelling curves for G-25 and G-50 gels and thecorresponding parity plots are shown in Figure 3. Thesecurves have been calculated using the overall diffusioncoefficient, for both gels.

Similarity between HDF and LDF. Simulationswith the HDF approach and the LDF approach givesimilar results, as can be seen from the parity plots inFigure 4. A constant value for the Sherwood number(Sh ) 2/3π2) can be used for describing the swellingdynamics of hydrogels. The parity plots indicate that

the HDF approach gives slightly higher swelling rates,mainly in the initial stage.

Results Using the Ogston Model. The Ogstonmodel for the effective diffusion coefficient was usedwith varying values for the size ratio (λ). Vonk22

reported the radius of the dextran polymer chains tobe rP ) 6.9 Å and that of glucose, the monomer units ofSephadex gels, to be rP ) 3.5 Å. The van der Waalsradius of water is rS ) 1.69 Å.27 This would indicate avalue for the size ratio of λ ) 4.1. We found a slightlyhigher value (λ ) 6) to give the best results.

Diffusion Coefficients. The estimated diffusioncoefficients listed in Table 3 are surprisingly large,compared to common molecular Maxwell-Stefan diffu-sion coefficients in water.

The large difference between the values for thehydrodynamic model and the Ogston model is causedby the different reference point for the effective diffusioncoefficient. The Ogston model leads to f ) 1 at infinitedilution (φ ) 0), whereas the hydrodynamic model yieldsf ) 1 at φ ) 0.382 and tends to infinity at infinitedilution (φ ) 0).

The diffusion coefficient does not seem to depend onthe degree of cross-linking. The swelling behavior ofboth types of Sephadex gels can be described withsatisfactory accuracy using the same diffusion coef-ficient.

The Kozeny-Carman equation can be used to predictthe behavior of the diffusion coefficient of water in thematrix according to the hydrodynamic model:

With a fiber diameter of df ) 1.38 nm,22 this yields D0) 3.3 × 10-9 m2/s, which has the same order ofmagnitude as the diffusion coefficient listed in Table 3.

Intraparticle Behavior. Simulations with the HDFapproach clearly indicate shrinking-core behavior. Thisis shown in Figure 5 for both the hydrodynamic modeland the Ogston model for the effective diffusion coef-ficient. The hydrodynamic model shows steeper pen-etrating solvent fronts than the Ogston model, leadingto a distinct later dissolution of the glassy core. This isshown in the simulations of two individual experimentsin Figure 6, in which the model curves for the outerradius and the boundary between the glassy core andthe swollen region are plotted.

The difference between the curves is caused by thefact that the hydrodynamic model predicts a very rapiddecline of the effective diffusion coefficient if the polymervolume fraction exceeds 80%. This results in a sharpincrease of the concentration gradient (dφ/dx) beyond avolume fraction of polymer of 80% for the hydrodynamicmodel, whereas the Ogston model leads to a decreasein the gradient beyond this point.

Table 2. Physical Properties of Sephadex G-25 and G-50

type G-25 G-50 units

particle size 100-300 100-300 µmbed volumea 5 10 mL/g of dry gelwater regainb 2.5 5.0 g/gwet density 1.13 1.07 g/mLmatrix dextrancross-linker epichlorohydrin% cross-linksc 3.6 0.68 %FH parameter ød 0.50swelling ratioe 4.2-5.6

(avg. 4.72)7.5-9.8

(avg. 8.65)V∞/V0

a The water regain is defined as the amount of water absorbedby 1 g of dry gel. b The bed volume is the volume of 1 g of dry gelwhen swollen in water. It includes approximately 40% voidbetween the (spherical) particles. c Fitted to the reported swellingratios. d Flory-Huggins parameter reported by Kang and San-dler.26 e Swelling ratio observed by Pratt.8

Table 3. Estimated Diluted Diffusion Coefficients for Sephadex Gels-Watera

G-25 data G-50 data combined

modeling approach D0 [m2/s] σ [%] D0 [m2/s] σ [%] D0 [m2/s] σ [%]

hydrodynamic model LDF 5.20 × 10-9 6.92 8.20 × 10-9 5.44 7.62 × 10-9 5.31HDF 5.32 × 10-9 6.98 8.00 × 10-9 5.39 7.50 × 10-9 5.42

Ogston model (λ ) 6) LDF 3.87 × 10-7 6.93 5.99 × 10-7 5.38 5.58 × 10-7 5.28HDF 3.74 × 10-7 6.63 5.71 × 10-7 5.45 5.34 × 10-7 5.37

a The results for the HDF approach are obtained with 80 grid cells in the radial direction.

σ ) [∑(Vmod - Vexp

V∞ - V0)2

N]1/2

(26)

(27)Deff =RTνS

df 2

80η(1 – φ)2

φ

D0 f (φ)


Lee11 reported that the thickness of the swollen regionoutside the core was approximately proportional to thesquare root of the swelling time. The GMS model showssimilar behavior for the dextran gels with both modelsfor the effective diffusion coefficient. However, sincethere are no data available for the solvent propagationvelocities, no clear distinction can be made between thetwo models at this point.

Conclusions

The dynamics of the swelling process of hydrogels aresuccessfully described on the basis of the generalized

Maxwell-Stefan description. The presented modelincludes a correction for the composition-dependenteffective diffusion coefficient. The model yields a dif-ferential equation that can be evaluated numerically bydiscretization in place and time. Application of thelinear driving force approach allows a simplification toan ordinary differential equation. For swelling in a puresolvent (the binary case), both approaches correspondclosely when the Sherwood number is chosen to beanalogous to the classical mass-transfer theories (Sh )2/3π2).

Figure 3. Two typical swelling curves (top) and corresponding parity plots (bottom). Left: Sephadex G-25 gel in water; model curvebased on the HDF and the hydrodynamic model for the effective diffusion coefficient (D0 ) 7.50 × 10-9 m2/s). Right: Sephadex G-50 gelin water; model curves based on the LDF and the Ogston model for the effective diffusion coefficient (D0 ) 5.58 × 10-7 m2/s).

Figure 4. Parity plots for the homogeneous driving force (HDF) and linear driving force (LDF) approaches. Left: the hydrodynamicmodel for the effective diffusion coefficient. Right: the Ogston model for the effective diffusion coefficient.


The advantages of the model presented in this workare as follows:

(1) The model is able to describe the experimentaldata reported by Pratt and Cooney8,9 very accurately,

whereas other models show an underestimation of theinitial swelling rate followed by an overestimation of theswelling rate in a later stage of the process.

(2) It requires pure-component properties only and

Figure 5. Concentration profiles in Sephadex G-25 (top) and G-50 (bottom) gel particles. The curves indicate the polymer volume fractionas a function of the dimensionless radius at different moments. The effective diffusion coefficient is evaluated with the hydrodynamicmodel (left) and with the Ogston model (right).

Figure 6. Solvent penetration as a function of the square root of time (xt) during the swelling of Sephadex G-25, (left) and G-50 (right)gels.


one diffusion coefficient for each pair of componentsinvolved. The diffusion coefficient does not seem todepend on the degree of swelling and can thereforesafely be assumed to depend solely on the solvent andpolymer chains.

(3) In principle, it is not limited to hydrogels with amoderate equilibrium swelling ratio. The model is alsoapplicable to nonlinear systems, provided that theeffective diffusion coefficient can be described correctly.

(4) In contrast to earlier models, the GMS modelpredicts the shrinking-core behavior. The penetrationof the solvent front according to the hydrodynamicmodel and the Ogston model corresponds qualitativelyto the behavior reported by Lee.11

(5) Analogous to the other GMS applications, theswelling model presented in this work can be extendedstraightforwardly to multicomponent systems. Thisshould allow description of swelling processes in thepresence of cosolvents and solutes.

The high value for the estimated diffusion coefficientis mainly due to the fact that it is based on the volumefraction rather than on the mole fraction. This diffusioncoefficient should, therefore, not be compared withvalues for common GMS molecular diffusion coefficients.

Acknowledgment

The authors gratefully acknowledge Prof. R. Taylorfrom Clarkson University for being so kind to providea copy of the M.Sc. thesis of C. F. Pratt. This researchwork is part of a project which is financially supportedby the Dutch Department of Economic Affairs in theframework of IOP MT Preventie.

Symbols

a ) specific interfacial area in the fiber bed [m2/m3]B ) fiber bed permeability [m2]dP ) particle diameter [m]df ) diameter of the polymer chain segments [m]Deff ) effective diffusion coefficient [m2/s]D0 ) diffusion coefficient at infinite dilution [m2/s]f ) relative diffusion coefficient (Deff/D0)Fdr ) driving force [N/mol]Ffr ) friction force [N/mol]Fo ) Fourier number (tD/r2)M ) molecular weight [g/mol]P ) pressure [Pa]r ) radial coordinate in the gel bead [m]rP ) radius of the gel bead [m]rP ) radius of the polymer chain segments [m]rS ) radius of the diffusing molecule [m]rW ) van der Waals radius of the diffusing molecule [m]r0 ) initial radius of the gel bead [m]R ) gas constant (8.314) [J/mol/K]T ) temperature [K]t ) time [s]u ) linear velocity [m/s]V ) volume of the gel bead [m3]x ) dimensionless radial coordinate inside the gel bead (r/

r0)X ) dimensionless radius of the gel bead (rP/r0)

Greek Symbols

φ ) volume fraction of polymer in the gel beadΦ ) network functionalityλ ) size ratio between polymer segment and diffusing

molecule (rP/rS)µ ) chemical potential [J/mol]ν ) partial molar volume [m3/mol]

θ ) dimensionless (Fourier) time (tD0/r02)

ø ) Flory-Huggins interaction parameter

Subscripts

exp ) experimental observationmod ) result from model calculationsi ) space index in the numerical procedure0 ) initial stateS ) solventP ) polymer∞ ) at equilibrium

Superscripts

FH ) Flory-Huggins contribution to the chemical potentialel ) elastic contribution to the chemical potential0 ) standard state

Appendix: Derivation of the Sherwood Number

The derivation of the Sherwood number (Sh) is basedon a comparison between the homogeneous solid-phasediffusion model (HSDM), based on Fick’s law, and thelinear driving force (LDF) model, which is based on thetwo-film theory by Lewis and Whitman.28

The dimensionless differential equation for the LDFapproach can be written as

where q represents the bulk concentration in the solidphase and qi represents the concentration at the inter-face. The dimensionless time is defined similarly to theFourier number (θ ) Dt/r2). For a constant interfaceconcentration (which is the case in contact with a puresolvent) and zero initial concentration, the analyticalsolution to the differential equation reads as follows:

For conduction of heat in solid bodies, Carslaw andJaeger29 derived an analytical expression for the tem-perature profile in a solid sphere with zero initialtemperature and constant surface temperature. Themathematical description of heat conduction is exactlyidentical with Fick’s law on diffusion. We can, therefore,adapt the analytical expression to the mass-transferproblem:

The bulk concentration in the sphere can be calcu-lated by taking the integral

which yields

By taking the analytical expressions for the HSDM andLDF equal, the Sherwood number can be calculated asa function of dimensionless time (θ):

dqdθ

) 32Sh(qi - q) (A.1)

qqi

) 1 - exp(- 32Shθ) (A.2)

q(R,θ)

qi) 1 +

2

πR∑n)1

∞ (-1)n

nsin(nπR) exp(-

θ

n2π2) (A.3)

q(θ) ) 3∫0

1R2q(θ,R) dR (A.4)

q(θ)

qi) 1 - ∑

n)1

∞ 6

n2π2exp(-n2π2θ) (A.5)


The Sherwood number is plotted as function of dimen-sionless time in Figure 7. It can be seen that theSherwood number corresponds to the penetration modelfor very short contact times17

whereas for longer contact times, the Sherwood numberreaches a steady-state value:

Literature Cited

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(8) Pratt, C. F. The Kinetics of Swelling of Dextran Gels inAqueous Protein Solutions. M.Sc. Thesis, Clarkson University,Potsdam, NY, 1972.

(9) Pratt, C. F.; Cooney, D. O. Kinetics of Swelling of DextranGels in Aqueous Protein Solutions. AIChE J. 1972, 19, 1049.

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(15) Crank, J. The Mathematics of Diffusion, 2nd ed.; ClarendonPress: Oxford, U.K., 1975.

(16) Komori, T.; Sakamoto, R. On Tanaka-Fillmore’s kineticsswelling of gels. Colloid Polym. Sci. 1989, 267, 179-183.

(17) Wesselingh, J. A.; Krishna R. Mass Transfer; Ellis Hor-wood: Chichester, U.K., 1990.

(18) Flory, P. J. Principles of Polymer Chemistry; CornellUniversity, Ithaca Press: New York, 1953.

(19) Hooper, H. H.; Baker, J. P.; Blanch, H. W.; Prausnitz, J.M. Swelling Equilibria for Positively Ionized PolyacrylamideHydrogels. Macromolecules 1990, 23, 1096.

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(22) Vonk, P. Diffusion of Large Molecules in Porous Structures.Ph.D. Thesis, Groningen University, Groningen, The Netherlands,1994.

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Received for review January 21, 1998Revised manuscript received May 19, 1998

Accepted May 20, 1998

IE9800389

Figure 7. Actual Sherwood number versus dimensionless time(θ) and the Sherwood numbers according to the penetration theoryand the steady-state value.

Sh )2

3θln(-6∑

n)1

∞ exp(n2π2θ)

n2π2 ) (A.6)

Sh ) 4/xπθ (A.7)

Sh ) (2/3)π2 (A.8)


generalized maxwell−stefan approach for swelling kinetics of dextran gels

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